Twist operators in higher dimensions

Hung, Ling-Yan; Myers, Robert C.; Smolkin, Michael

doi:10.1007/jhep10(2014)178

Cited by 113 publications

(239 citation statements)

References 104 publications

(242 reference statements)

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“…Interestingly, C T arises from a pre-factor, which is the derivative of the scaling dimension of a twist operator with respect to the Rényi index. This derivative was recently shown to be π 3 C T /24 for general CFTs [46] -see appendix C for further comments. This simple expression (14) is also shown in Fig.…”

Section: Discussionmentioning

confidence: 90%

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Universality of Corner Entanglement in Conformal Field Theories

2015

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We study the contribution to the entanglement entropy of (2+1)-dimensional conformal field theories coming from a sharp corner in the entangling surface. This contribution is encoded in a function a(θ) of the corner opening angle, and was recently proposed as a measure of the degrees of freedom in the underlying CFT. We show that the ratio a(θ)/CT , where CT is the central charge in the stress tensor correlator, is an almost universal quantity for a broad class of theories including various higher-curvature holographic models, free scalars and fermions, and Wilson-Fisher fixed points of the O(N ) models with N = 1, 2, 3. Strikingly, the agreement between these different theories becomes exact in the limit θ → π, where the entangling surface approaches a smooth curve. We thus conjecture that the corresponding ratio is universal for general CFTs in three dimensions.Many interacting gapless quantum systems do not possess simple particle-like excitations, making it difficult to quantify their effective number of degrees of freedom (dof) at low-energy. Conformal field theories (CFTs) constitute an important example. For CFTs in two spacetime dimensions (2d), the Virasoro central charge is a good measure of the dof. It appears in many quantities, such as the thermal free energy and the entanglement entropy (EE), and decreases under renormalization group (RG) flow [1]. In higher dimensions, the concept of quantum entanglement is emerging as a fundamental diagnostic for such measures [2,3]. E.g., it was instrumental in finding an analogous RG monotone for 3d CFTs, with the EE of a disk-shaped region [4]. We shall study another measure of recent interest [5][6][7][8][9][10][11][12][13][14][15][16]: the coefficient capturing the contribution of sharp corners to spatial entanglement.In the context of quantum field theory, the EE is defined for a spatial region V as: S = −Tr (ρ V ln ρ V ), where ρ V is the reduced density matrix produced by integrating out the dof in the complementary region V . In the groundstate of a 3d CFT, the EE takes the form:where δ is a short-distance cutoff, e.g., the lattice spacing, and , a length scale associated with the size of V . The first, 'area law', term depends on the UV regulator and scales with the size of the boundary. The second one appears only when V has a sharp corner with opening angle θ ∈ [0, 2π), Fig. 1. Crucially, a(θ) is a regulator independent coefficient that characterizes the underlying CFT. It is positive and satisfies a(2π − θ) = a(θ) [5], and behaves as follows:in the limits of a nearly smooth entangling surface and a very sharp corner, respectively. It has been studied for a variety of systems: free scalars and fermions [5][6][7], interacting scalar theories via numerical simulations [8][9][10], Lifshitz quantum critical points [11], and holographicFIG. 1: a) An entangling region V of size with a corner; b) The holographic entangling surface γ for a region on the boundary of AdS4 with a corner.models [12]. The results suggest that a(θ) is an effective measure of the do...

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Section: Discussionmentioning

confidence: 90%

“…Hence both the b 1 and b 2 terms in the correlator (C2) make the same contribution to the logarithmic term up to an overall factor. Following [37], we interpret α 2 n as the scaling dimension of the twist operator and hence we can apply the recent result of [46] ∂ n (α…”

Section: Appendix B: Field Theory Calculations Of σmentioning

confidence: 99%

Universality of Corner Entanglement in Conformal Field Theories

2015

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“…To use (2.26) we need the connected correlator T µν HĤ c . Since the Hamiltonian is conserved and hyperbolic space is maximally symmetric, the correlator is insensitive to where the operators are inserted, and therefore it is constant on H. In particular, it was shown in [48,49] that…”

Section: Geometric Perturbationsmentioning

confidence: 99%

Entanglement entropy: a perturbative calculation

Rosenhaus

Smolkin

2014

J. High Energ. Phys.

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104

192

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We provide a framework for a perturbative evaluation of the reduced density matrix. The method is based on a path integral in the analytically continued spacetime. It suggests an alternative to the holographic and 'standard' replica trick calculations of entanglement entropy. We implement this method within solvable field theory examples to evaluate leading order corrections induced by small perturbations in the geometry of the background and entangling surface. Our findings are in accord with Solodukhin's formula for the universal term of entanglement entropy for four dimensional CFTs.

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“…As in [14,15] we would like to focus on the ratio of the expectation value of the stress tensor to that of the surface operator alone. Simply from dimensional analysis, it is expected that the leading term of the OPE contains a 1/ǫ d divergence as ǫ → 0, and that the form of the leading term is controlled by a single coefficient for a conformal field theory [16][17][18]. It is known that the leading term does not actually take part in controlling the transformation of the surface operator under scaling or translation [16].…”

Section: Jhep06(2015)087mentioning

confidence: 99%

OPE of the stress tensors and surface operators

Huang

Hung

Lin

2015

J. High Energ. Phys.

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We demonstrate that the divergent terms in the OPE of a stress tensor and a line (co-dimension two) operator of general shape in three dimensional spacetime cannot be constructed only from local geometric data depending only on the shape of the line. We verify this holographically for Wilson line operators or equivalently the twist operator corresponding to computing the entanglement entropy using the Ryu-Takayanagi formula. We discuss possible implications of this result.

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Twist operators in higher dimensions

Cited by 113 publications

References 104 publications

Universality of Corner Entanglement in Conformal Field Theories

Universality of Corner Entanglement in Conformal Field Theories

Entanglement entropy: a perturbative calculation

OPE of the stress tensors and surface operators

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